For every integer $g$, we construct a $2$-solvable and $2$-bipolar knot whose topological $4$-genus is greater than $g$. Note that $2$-solvable knots are in particular algebraically slice and have vanishing Casson–Gordon obstructions. Similarly all known smooth $4$-genus bounds from gauge theory and Floer homology vanish for $2$-bipolar knots. Moreover, our knots bound smoothly embedded height four gropes in $D^4$, an a priori stronger condition than being $2$-solvable. We use new lower bounds for the $4$-genus arising from $L^{(2})$-signature defects associated to meta-metabelian representations of the fundamental group.

中文翻译：

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具有四大类的两个可解和两个双极结

对于每个整数$ g $，我们构造一个可解决的$ 2 $和一个$ 2 $双极结，其拓扑$ 4 $属大于$ g $。请注意，$ 2 $可解结尤其是代数切片，并具有消失的Casson–Gordon障碍物。类似地，从轨距理论和Floer同源性得出的所有已知的平滑4元属界都以2元双极结消失。此外，我们的节点将高度为4的摸索平滑地嵌入到$ D ^ 4 $中，这是先验条件，比可解决$ 2 $的条件强。我们对$ 4 $属使用新的下界，该下界是由与基本组的元metabelian表示相关的$ L ^ {（2}）$签名缺陷引起的。